Finite Group Databasebeta

← #10604629

Subgroups — #10604629

Order 640.

208 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#66062C211yesmaximal, characteristic
2#65690C211yesmaximal, characteristic
3#65684C211yesmaximal, characteristic
4#64926C214yesmaximal
5C5×C4⋊Q8C411yescharacteristic, complemented
6#4092C2212yes—
7Q8×Dic5C412yescomplemented
8#4043—21no—
9#4038—21no—
10#4036—21no—
11#4033—21no—
12C42×D5C2211yescharacteristic
13#4015C411yescharacteristic, complemented
14(C2×AGL(1,𝔽5))⋊C4—21no—
15(C2×AGL(1,𝔽5))⋊C4—21no—
16(C2×AGL(1,𝔽5))⋊C4C2212yes—
17#4004—21no—
18#4003C2212yes—
19#4002—21no—
20#1515—51nomaximal, Sylow
21C22×AGL(1,𝔽5)—22no—
22C2×C5×Q8C2×C412yes—
23C2×C4×D5D411yescharacteristic
24C2×C4×D5D411yescharacteristic
25C2×C4×D5C2311yescharacteristic
26C2×Dic10—21no—
27C2×#170—22no—
28C5×C4⋊C4—21no—
29C5×C4⋊C4—21no—
30C42×C5C2×C411yescharacteristic
31#685—22no—
32C4⋊Dic5—21no—
33C4⋊Dic5—22no—
34C4⋊Dic5C2×C412yes—
35#683—22no—
36C4×Dic5—22no—
37C4×Dic5C2×C411yescharacteristic
38C4×Dic5D411yescharacteristic
39C4×Dic5D411yescharacteristic
40#533—51no—
41#432—51no—
42#383—51no—
43#328—54no—
44C22×D5C2×D411yescharacteristic
45C2×AGL(1,𝔽5)—42no—
46C2×AGL(1,𝔽5)—42no—
47C2×AGL(1,𝔽5)—42no—
48C5×Q8—42no—
49C2×C4×C5—22no—
50C2×C4×C5C22⋊C411yescharacteristic
51C2×C4×C5C22⋊C411yescharacteristic
52C2×C4×C5C22×C411yescharacteristic
53C2×Dic5—22no—
54C2×Dic5C22⋊C411yescharacteristic
55C2×Dic5C22⋊C411yescharacteristic
56C2×Dic5C2×D411yescharacteristic
57C4×D5—22no—
58C4×D5—41no—
59C4×D5—41no—
60Dic10—42no—
61#170—42no—
62C4⋊Q8—51no—
63C4⋊Q8AGL(1,𝔽5)11yescharacteristic, complemented
64#125—101no—
65C22⋊Q8—52no—
66C22⋊Q8—101no—
67C4×Q8—52no—
68C4×Q8—101no—
69#117—101no—
70C2×C42—51no—
71C4⋊C8—101no—
72C22⋊C8—52no—
73#97—101no—
74#95—52no—
75#95—101no—
76#95—101no—
77C22×C5C2×C22⋊C411yescharacteristic
78C2×D5—21no—
79C2×D5—21no—
80C2×D5#13712yes—
81AGL(1,𝔽5)—82no—
82C4×C5—21no—
83C4×C5—21no—
84C4×C5—21no—
85C4×C5—42no—
86Dic5—21no—
87Dic5—21no—
88Dic5—21no—
89Dic5—42no—
90C2×Q8—101no—
91C2×Q8C2×AGL(1,𝔽5)12yes—
92C22×C4—51no—
93C22×C4—51no—
94C22×C4—51no—
95C22×C4—102no—
96C2×C8—102no—
97C4⋊C4—21no—
98C4⋊C4—21no—
99C4⋊C4—52no—
100C4⋊C4—102no—
101C4⋊C4—102no—
102C4⋊C4—101no—
103C22⋊C4—102no—
104C42—51no—
105C42—51no—
106C42—51no—
107C42—102no—
108C42C2×AGL(1,𝔽5)11yescharacteristic
109C2×C5#41011yescharacteristic
110C2×C5#41911yescharacteristic
111C2×C5#42111yescharacteristic
112D5—22no—
113C23—51no—
114Q8—42no—
115Q8—202no—
116C2×C4—22no—
117C2×C4—51no—
118C2×C4—51no—
119C2×C4—51no—
120C2×C4—102no—
121C2×C4—102no—
122C2×C4—201no—
123C2×C4—201no—
124C2×C4—202no—
125C2×C4—202no—
126C2×C4—202no—
127C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
128C2×C4C22⋊AGL(1,𝔽5)11yescharacteristic
129C2×C4C22×AGL(1,𝔽5)11yescharacteristic
130C8—202no—
131C5#151511yescharacteristic, complemented, Sylow
132C22—52no—
133C22—101no—
134C22—101no—
135C22C2×C22⋊AGL(1,𝔽5)11yescharacteristic
136C4—21no—
137C4—21no—
138C4—21no—
139C4—42no—
140C4—101no—
141C4—101no—
142C4—101no—
143C4—202no—
144C4—402no—
145C2—102no—
146C2#6577111yescharacteristic
147C2#6578211yescharacteristic
148C2#6578411yescharacteristic
subgroupG/H
centerC22C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C4
FrattiniC2×C4C22×AGL(1,𝔽5)
FittingC5×C4⋊Q8C4
socleC22×C5C2×C22⋊C4
prime radicalC22×D5C2×D4
cosocleC2×C4×D5C23
hypercenterC4⋊Q8AGL(1,𝔽5)
hypocenterC5#1515
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. C5×C4⋊Q8C4semidirect
7. Q8×Dic5C4semidirect
13. #4015C4semidirect
63. C4⋊Q8AGL(1,𝔽5)semidirect
131. C5#1515semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#660622
#656902
#656842
#649262
#15155